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Keywords:
singular boundary value problem; positive solution; upper and lower solution
singular boundary value problem; positive solution; upper and lower solution
Summary:
This paper studies the existence of solutions to the singular boundary value problem \[ \left\rbrace \begin{array}{ll}-u^{\prime \prime }=g(t,u)+h(t,u),\quad t\in (0,1) , u(0)=0=u(1), \end{array}\right.\] where $g\:(0,1)\times (0,\infty )\rightarrow \mathbb{R}$ and $h\:(0,1)\times [0,\infty )\rightarrow [0,\infty )$ are continuous. So our nonlinearity may be singular at $t=0,1$ and $u=0$ and, moreover, may change sign. The approach is based on an approximation method together with the theory of upper and lower solutions.
References:
[1] R. P. Agarwal, D. O’Regan: Singular Differential and Integral Equations with Applications. Kluwer Academic Publishers, Dordrecht, 2003. MR 2011127
[2] P. Habets, F. Zanolin: Upper and lower solutions for a generalized Emden-Fower equation. J. Math. Anal. Appl. 181 (1994), 684–700. DOI 10.1006/jmaa.1994.1052 | MR 1264540
[3] D. O’Regan: Theory of Singular Boundary Value Problems. World Scientific, Singapore, 1994. MR 1286741
[4] H. Lü, D. O’Regan, and R. P. Agarwal: An Approximation Approach to Eigenvalue Intervals for Singular Boundary Value Problems with Sign Changing Nonlinearities. to appear.
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